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Removing the Stiffness of Elastic Force from the Immersed Boundary Method for the 2D Stokes Equations
, 2008
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An implicit immersed boundary method for threedimensional . . .
 JOURNAL OF COMPUTATIONAL PHYSICS
, 2009
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A Multirate Time Integrator for Regularized Stokeslets
"... The method of regularized Stokeslets is a numerical approach to approximating solutions of fluidstructure interaction problems in the Stokes regime. Regularized Stokeslets are fundamental solutions to the Stokes equations with a regularized pointforce term that are used to represent forces generat ..."
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The method of regularized Stokeslets is a numerical approach to approximating solutions of fluidstructure interaction problems in the Stokes regime. Regularized Stokeslets are fundamental solutions to the Stokes equations with a regularized pointforce term that are used to represent forces generated by rigid or elastic object interacting with the fluid. Due to the linearity of the Stokes equations, the velocity at any point in the fluid can be computed by summing the contributions of regularized Stokeslets, and the time evolution of positions can be computed using standard methods for ordinary differential equations. Rigid or elastic objects in the flow are usually treated as immersed boundaries represented by a collection of regularized Stokeslets coupled together by virtual springs which determine the forces exerted by the boundary in the fluid. For problems with boundaries modeled by springs with large spring constants, the resulting ordinary differential equations become stiff, and hence the time step for explicit time integration methods is severely constrained. Unfortunately, the
A Velocity Decomposition Approach for Moving Interfaces in Viscous Fluids
"... We present a secondorder accurate method for computing the coupled motion of a viscous fluid and an elastic material interface with zero thickness. The fluid flow is described by the NavierStokes equations, with a singular force due to the stretching of the moving interface. We decompose the veloc ..."
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We present a secondorder accurate method for computing the coupled motion of a viscous fluid and an elastic material interface with zero thickness. The fluid flow is described by the NavierStokes equations, with a singular force due to the stretching of the moving interface. We decompose the velocity into a “Stokes ” part and a “regular ” part. The first part is determined by the Stokes equations and the singular interfacial force. The Stokes solution is obtained using the immersed interface method, which gives secondorder accurate values by incorporating known jumps for the solution and its derivatives into a finite difference method. The regular part of the velocity is given by the NavierStokes equations with a body force resulting from the Stokes part. The regular velocity is obtained using a timestepping method that combines the semiLagrangian method with the backward difference formula. Because the body force is continuous, jump conditions are not necessary. For problems with stiff boundary forces, the decomposition approach can be combined with fractional timestepping, using a smaller time step to advance the interface quickly by Stokes flow, with the velocity computed using boundary integrals. The small time steps maintain numerical stability, while the overall solution is updated on a larger time step to reduce computational cost.
A PARTIALLY IMPLICIT HYBRID METHOD FOR COMPUTING INTERFACE MOTION IN STOKES FLOW
"... We present a partially implicit hybrid method for simulating the motion of a stiff interface immersed in Stokes flow, in free space or in a rectangular domain with boundary conditions. The implicit time integration is based on the smallscale decomposition approach and does not require the iterativ ..."
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We present a partially implicit hybrid method for simulating the motion of a stiff interface immersed in Stokes flow, in free space or in a rectangular domain with boundary conditions. The implicit time integration is based on the smallscale decomposition approach and does not require the iterative solution of a system of nonlinear equations. Firstorder and secondorder versions of the timestepping method are derived systematically, and numerical results indicate that both methods are substantially more stable than explicit methods. At each time level, the Stokes equations are solved using a hybrid approach combining nearly singular integrals on a band of mesh points near the interface and a meshbased solver. The solutions are secondorder accurate in space and preserve the jump discontinuities across the interface. Finally, the hybrid method can be used as an alternative to adaptive mesh refinement to resolve boundary layers that are frequently present around a stiff immersed interface.
An Efficient Parallel Immersed Boundary Algorithm using a PseudoCompressible Fluid Solver
, 2013
"... We propose an efficient algorithm for the immersed boundary method on distributedmemory architectures, with the computational complexity of a completely explicit method and excellent parallel scaling. The algorithm utilizes the pseudocompressibility method recently proposed by Guermond and Minev t ..."
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We propose an efficient algorithm for the immersed boundary method on distributedmemory architectures, with the computational complexity of a completely explicit method and excellent parallel scaling. The algorithm utilizes the pseudocompressibility method recently proposed by Guermond and Minev that uses a directional splitting strategy to discretize the incompressible NavierStokes equations, thereby reducing the linear systems to a series of onedimensional tridiagonal systems. We perform numerical simulations of several fluidstructure interaction problems in two and three dimensions and study the accuracy and convergence rates of the proposed algorithm. For these problems, we compare the proposed algorithm against other secondorder projectionbased fluid solvers. Lastly, the strong and weak scaling properties of the proposed algorithm are investigated.
IMMERSED BOUNDARY METHOD FOR VARIABLE VISCOSITY AND VARIABLE DENSITY PROBLEMS USING FAST CONSTANTCOEFFICIENT LINEAR SOLVERS I: NUMERICAL METHOD AND RESULTS
"... Abstract. We present a general variable viscosity and variable density immersed boundary method that is firstorder accurate in the variable density case and, for problems possessing sufficient regularity, secondorder accurate in the constant density case. The viscosity and density are considered m ..."
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Abstract. We present a general variable viscosity and variable density immersed boundary method that is firstorder accurate in the variable density case and, for problems possessing sufficient regularity, secondorder accurate in the constant density case. The viscosity and density are considered material properties and are defined by a dynamically updated tesselation. Empirical convergence rates are reported for a test problem of a twodimensional viscoelastic shell with spatially varying material properties. The reduction to firstorder accuracy in the variable density case can be avoided by using an iterative scheme, although this approach may not be efficient enough for practical use. In our timestepping scheme, both the inertial and viscous terms are split into two parts: a constantcoefficient part that is treated implicitly, and a variablecoefficient part that is treated explicitly. This splitting allows the resulting equations to be solved efficiently using fast constantcoefficient linear solvers, and in this work, we use solvers based on the Fast Fourier transform (FFT). As an application of this method, we perform fully threedimensional, twophase simulations of red blood cells accounting for variable viscosity and variable density. We study the behavior of red cells during shear flow and during capillary flow. Key words. immersed boundary method, variable viscosity, red blood cells AMS subject classifications. 65M06, 76D05, 76Z05
Unconditionally Energy Stable Immersed Boundary Method with Application to Vesicle Dynamics
"... Abstract. We develop an unconditionally energy stable immersed boundary method, and apply it to simulate 2D vesicle dynamics. We adopt a semiimplicit boundary forcing approach, where the stretching factor used in the forcing term can be computed from the derived evolutional equation. By using the p ..."
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Abstract. We develop an unconditionally energy stable immersed boundary method, and apply it to simulate 2D vesicle dynamics. We adopt a semiimplicit boundary forcing approach, where the stretching factor used in the forcing term can be computed from the derived evolutional equation. By using the projection method to solve the fluid equations, the pressure is decoupled and we have a symmetric positive definite system that can be solved efficiently. The method can be shown to be unconditionally stable, in the sense that the total energy is decreasing. A resulting modification benefits from this improved numerical stability, as the time step size can be significantly increased (the severe time step restriction in an explicit boundary forcing scheme is avoided). As an application, we use our scheme to simulate vesicle dynamics in NavierStokes flow.
Viscoelastic Immersed Boundary Methods for Zero Reynolds Number Flow
"... Abstract. The immersed boundary method has been extensively used to simulate the motion of elastic structures immersed in a viscous fluid. For some applications, such as modeling biological materials, capturing internal boundary viscosity is important. We present numerical methods for simulating Kel ..."
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Abstract. The immersed boundary method has been extensively used to simulate the motion of elastic structures immersed in a viscous fluid. For some applications, such as modeling biological materials, capturing internal boundary viscosity is important. We present numerical methods for simulating KelvinVoigt and standard linear viscoelastic structures immersed in zero Reynolds number flow. We find that the explicit time immersed boundary update is unconditionally unstable above a critical boundary to fluid viscosity ratio for a KelvinVoigt material. We also show there is a severe time step restriction when simulating a standard linear boundary with a small relaxation time scale using the same explicit update. A stable implicit method is presented to overcome these computation challenges. AMS subject classifications: 65M06,65M12,74F10,76D07,76M20 1
SIMULATING FLEXIBLE FIBER SUSPENSIONS USING A SCALABLE IMMERSED BOUNDARY ALGORITHM
"... We present an approach for numerically simulating the dynamics of flexible fibers in a threedimensional shear flow using a scalable immersed boundary (IB) algorithm based on Guermond and Minev’s pseudocompressible fluid solver. The fibers are treated as onedimensional Kirchhoff rods that resist ..."
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We present an approach for numerically simulating the dynamics of flexible fibers in a threedimensional shear flow using a scalable immersed boundary (IB) algorithm based on Guermond and Minev’s pseudocompressible fluid solver. The fibers are treated as onedimensional Kirchhoff rods that resist stretching, bending, and twisting, within the generalized IB framework. We perform a careful numerical comparison against experiments on single fibers performed by S. G. Mason and coworkers, who categorized the fiber dynamics into several distinct orbit classes. We show that the orbit class may be determined using a single dimensionless parameter for low Reynolds flows. Lastly, we simulate dilute suspensions containing up to hundreds of fibers using a distributedmemory computer cluster. These simulations serve as a stepping stone for studying more complex suspension dynamics including nondilute suspensions and aggregation of fibers (also known as flocculation).